COMPLETE PARTITE SUBGRAPHS IN DENSE HYPERGRAPHS
VOJT ˇECH R ¨ODL AND MATHIAS SCHACHT
Abstract. For a given r-uniform hypergraph F we study the largest blow-up of F which can be guaranteed in every large r-uniform hypergraph with many copies of F . For graphs this problem was addressed by Nikiforov, who proved that every n-vertex graph that contains Ω(n`) copies of the complete graph K`
must contain a complete `-partite graph with Ω(log n) vertices in each class.
We give another proof of Nikiforov’s result, make very small progress towards that problem for hypergraphs, and consider a Ramsey-type problem related to a conjecture of Erd˝os and Hajnal.
1. Introduction
We say a subhypergraph B of an r-uniform hypergraph H is a blow-up of an r- uniform hypergraph F if it is isomorphic to a hypergraph which is obtained from F by replacing every vertex v of F by an independent set Uv and for every edge e = {v1, . . . , vr} of F we add a complete r-partite r-uniform hypergraph spanned by Uv1, . . . , Uvr. We say a blow-up of F is a t-blow-up if |Uv| ≥ t for every v ∈ V (F ).
This note concerns the size of the largest blow-up of a given hypergraph F in a large hypergraph H which contains many copies of F .
One of the earliest results concerning blow-ups in dense graphs is due to K¨ovari, S´os, and Tur´an [12]. In fact, it follows from their result that for sufficiently large n every n-vertex graph with cn2 edges contains a t-blow-up of K2 (the graph con- sisting of a single edge) where t = γ log n and γ > 0 only depends on c (we write log for the logarithm with base 2 here). Indeed such a result appeared implicitly already in the work of Erd˝os and Stone [8]. This was generalized by Erd˝os [4] to r-uniform hypergraphs r ≥ 3. Erd˝os showed that for sufficiently large n every r- uniform hypergraph on n vertices with cnredges contains a t-blow-up of Kr(r)(the r-uniform hypergraph consisting of a single edge) where
t ≥ γ(log n)r−11 (1)
and γ > 0 only depends on c.
Note that the result of Erd˝os implies the following: if an n-vertex r-uniform hy- pergraph H contains at least cn`copies of K`(r), the complete r-uniform hypergraph
Key words and phrases. K¨ovari-S´os-Tur´an theorem, Erd˝os-Hajnal conjecture, partite hyper- graphs, blow-ups.
First author was supported by NSF grant DMS 0800070 and by an Emory University research grant.
Second author was supported through the Heisenberg-Programme of the Deutsche Forschungs- gemeinschaft (DFG Grant SCHA 1263/4-1).
Part of this work was carried out at the Institute of Advanced Studies in Princeton during the special semester on Arithmetic Combinatorics in Fall 2007 organized by J. Bourgain and V. Vu.
1
on ` vertices, then H contains a t-blow-up of K`(r) for some t ≥ γ(log n)`−11 .
For that we consider an auxiliary `-uniform hypergraph on the same vertex set, where every edge corresponds to a copy of K`(r). Applying the result of Erd˝os to to this auxiliary hypergraph yields a t-blow-up of K`(`), which corresponds to a t-blow-up of K`(r) in H.
However, applying (1) here seems to be wasteful. In fact, for graphs Nikiforov [15]
obtained a much better bound by an elegant application of the K¨ovari-S´os-Tur´an theorem.
Theorem 1 (Nikiforov [15]). For every ` ≥ 3 and c > 0 there exist γ > 0 and n0
such that every graph G = (V, E) on n ≥ n0 vertices which contains cn` copies of K` contains a t-blow-up of K` for some t ≥ γ log n.
We obtained a different proof of Theorem 1, which we present in Section 2.
A simple probabilistic argument shows that up to the constant γ Theorem 1 is optimal. In view (1) and Theorem 1the following problem for hypergraphs seems plausible.
Problem 2. For all integers ` > r > 2, every r-uniform hypergraph F with V (F ) = [`], and every c > 0 does there exist γ > 0 and n0 such that the following holds? If an r-uniform hypergraph H = (V, E) on n = |V | ≥ n0 vertices contains at least cn` copies of F , then H contains a t-blow-up of F for some
t ≥ γ(log n)r−11 .
Clearly, it would suffice to solve Problem 2 for F = K`(r) and the results of K¨ovari, S´os, and Tur´an [12], Erd˝os [4], and Nikiforov [15] answer the problem in an affirmative way for r = 2 and for ` = r > 2. A Ramsey-type version of Problem 2 for r = 3 was solved by Conlon, Fox, and Sudakov [3]. Here, we make very little progress on this problem and answer it affirmatively for some special hypergraphs (see Section4).
Perhaps the first open case of Problem2 is when F = K4−(3), i.e., the 3-uniform hypergraph on four vertices with three edges. For that case one straightforward approach is the following: First one applies an averaging argument to H to locate a set T of Ω(√
log n) vertices such that their joint link graph GT (i.e., the graph consisting of all those pairs of vertices that form a hyperedge in H with every vertex in T ) contains n3/ exp(O(√
log n)) triangles. Then one would have to show that GT contains an Ω(√
log n)-blow-up of the triangle K3. It seems plausible that such a statement might be true in general and we put forward the following problem.
Problem 3. For every C > 0 does there exist γ > 0 and n0such that the following holds? If G is a graph on n ≥ n0 vertices which contains at least n3/ exp(C√
log n) copies of K3, then G contains a t-blow-up of K3 for some t ≥ γ√
log n.
If one would like to extend the strategy discussed above to move from K4−(3) to the K4(3), then one would consider those hyperedges e of H which are supported by a triangle in GT. In particular, every such edge e forms a K4(3) with every vertex in T and if one could show that those edges still span a t-blow-up of an hyperedge for t = Ω(√
log n), then one would solve Problem 2for the K4(3).
Such an approach leads to the study of relatively dense 3-uniform hypergraphs with respect to an underlying graph. For a graph G and a 3-uniform hypergraph H on the same vertex set V we say G underlies H if the three vertices of every edge in H span a triangle in G. By d(H|G) we denote the relative density of H with respect to G defined to be the number of edges of H divided by the number of triangles in G (we only consider graphs containing at least one triangle here). Our next contribution is the following result.
Theorem 4. For every K > 0 there exists some β > 0 such that for every c > 0 there exists n0 such that the following holds.
If an n-vertex graph G with at least cn3 triangles underlies a 3-uniform hyper- graph H and d(H|G) ≥ (1 − β), then H contains a K√
log n-blow-up of one triplet, i.e., H contains a complete 3-partite, 3-uniform hypergraph with vertex classes of size at least K√
log n.
Note that Theorem 4 is an extension of the result of Erd˝os [4] for 3-uniform hypergraphs. In fact, for G being the complete graph it was proved by Erd˝os.
However, Theorem 4 shows that K → ∞ as β → 0 independent of c. Moreover, letting H be a random selection of (1 − β)cn3 triangles from an arbitrary graph with cn3 triangles shows that Theorem4 is best possible.
Theorem4has some interesting consequences related to a well known problem of Erd˝os and Hajnal from [6] (see also [5, p. 95]). Those authors conjectured that for every graph F there exists an ε > 0 such that for sufficiently large n every n-vertex graph G that does not contain an induced copy of F , must have the property that either G or its complement must contain a clique Kt for t = nε.
Despite a lot of effort this conjecture is still wide open. On the other hand, the variant of the conjecture when the clique Kt is replaces by a bipartite graph Kt,t
can be shown fairly easily (see, e.g. [5, Proposition 1]). A standard application of the regularity method for hypergraphs together with Theorem4 yields a first step towards the partite version of the Erd˝os-Hajnal problem for hypergraphs.
Corollary 5. For every 3-uniform hypergraph F and every constant K > 0 there exists n0 such that the following holds. If a 3-uniform hypergraph H on n ≥ n0
vertices contains no induced copy of F , then either H or its complement contains a t-blow-up of a single edge K3(3) for some t ≥ K√
log n.
We remark that Conlon, Fox, and Sudakov [2] recently showed that the con- stant K in Corollary5can be replaced by (log n)δ for some sufficiently small con- stant δ > 0 depending on F .
1.1. Organization. We begin in the next section with an alternative proof of Nikiforov’s theorem, Theorem1. In Section3we prove Theorem4and we sketch the proof of Corollary5. We close with a few positive examples concerning Problem2 in Section4.
2. Complete partite subgraphs in graphs with many cliques Nikiforov’s elegant proof of Theorem1 appears to be tailored for graphs. The following proof is somewhat different and we hoped that it may pave the way for extensions of Theorem1. Similarly, as in the original proof we will make use of the K¨ovari-S´os-Tur´an theorem.
Theorem 6 (K¨ovari, S´os & Tur´an 1954). If G = (A ˙∪B, E) is a bipartite graph and for some integer s and t we have
(t − 1)|A|
s
< |B||E|/|B|
s
(2) then G contains a complete bipartite graph Ks,t with the vertex class of size s contained in A and the vertex class of size t contained in B.
In particular, for every c > 0 there exists γKST > 0 and n0 such that every bipartite graph with n ≥ n0 vertices in each partition class and with at least cn2 edges contains a copy of Kt,t for some t ≥ γKSTlog n. We will use the following notation. Let G = (V, E) be a graph. For two disjoint sets U and W ⊆ V we write EG(U, W ) for the set of edges in G with one vertex in U and one vertex in W and set eG(U, W ) = |EG(U, W )|. For a vertex v ∈ V , we denote by NG(v) the set of neighbours of v in G, i.e., NG(v) = {u ∈ V : uv ∈ E}. For two vertices u, v ∈ V we denote by NG(u, v) the joint neighbourhood NG(u) ∩ NG(v).
More generally, for a set U ⊆ V we write NG(U ) forT
u∈UNG(u).
We continue with an alternative proof of Nikiforov’s theorem.
Proof of Theorem 1. First we will only consider the case ` = 3. We comment on the general case at the end of the proof. A simple averaging argument shows that we may assume without loss of generality that G is a balanced tripartite graph with vertex set X ˙∪Y ˙∪Z of size |X| = |Y | = |Z| = n.
For given c > 0 let γKST > 0 be the constant guaranteed by Theorem 6 such that every bipartite graph with density c/4 and vertex classes of size at least m (for sufficiently large m) contains a Kt0,t0 for t0≥ γKSTlog m. We set
γ = min
c
20 log(8/c),γKST 2
. (3)
For sufficiently large n let G = (X ˙∪Y ˙∪Z, EG) be a tripartite graph with vertex sets of size |X| = |Y | = |Z| = n which contains cn3 triangles. We fix auxiliary constants
s =
log(n) 2 log(4/c)
and r =
c log(n) 16 log(8/c)
<cs
8 . (4)
In a first step we remove all edges from EG(X, Y ) which are contained in only a few triangles. More precisely, we remove all edges xy with
NG(x, y) < cn/2 .
This way we destroy at most cn3/2 triangles in G and, hence, at least cn3/2 triangles are left after this deletion in the resulting graph. Let G1be the resulting graph. Consequently, at least cn2/2 edges between X and Y remain in G1. We select a subset X0⊆ X with at least s elements such that for
Y0= Y ∩ NG1(X0) we have
|Y0| ≥c 4
s n
(4)
>√ n .
In fact, such a set X0 exists since in view of the fact that eG1(X, Y ) ≥ cn2/2 we have
X
y∈Y
|NG1(y) ∩ X|
s
≥ ncn/2 s
≥c 4
s
nn s
.
We denote the induced subgraph of G1 on X0∪Y˙ 0∪Z by G˙ 0. Note that by the definition of Y0 the bipartite subgraph G0[X0, Y0] is complete and G0[X0, Y0] ⊆ G1[X0, Y0]. Consequently, every edge xy ∈ EG0(X0, Y0) satisfies
|NG0(x, y)| = |NG1(x, y)| = |NG(x, y)| ≥ cn/2 and, hence, the graph G0 contains at least c|X0||Y0|n/2 triangles.
Next we repeat the same argument for edges between X0 and Z. For that we remove all edges xz ∈ EG0(X0, Z) for which NG0(x, z) < c|Y0|/4. Similarly to the argument above, at least c|X0||Y0|n/4 triangles remain and, hence, c|X0|n/4 edges will be left between X0 and Z after this deletion. Let G2⊆ G0 be the graph consisting of the remaining edges. As a result we have
X
z∈Z
|NG2(z) ∩ X0| r
≥ nc|X0|/4 r
≥c 8
r n|X0|
r
and, hence, there exists a subset X00⊆ X0 of size r such that the joint neighbour- hood Z0= Z ∩T
x∈X00NG0(x) satisfies
|Z0| ≥c 8
r
n(4)> √ n .
Moreover, since every edge xz between X00 and Z0 is contained in at least c|Y0|/4 triangles the induced subgraph G00= G[X00, Y0, Z0] contains at least c|X00||Y0||Z0|/4 triangles and eG00(Y0, Z0) ≥ c|Y0||Z0|/4. Owing to Theorem 6 we have G00[Y0, Z0] contains a Kt0,t0 for
t0≥ γKSTlog(√ n) . Consequently, G contains a t-blow-up of K3 for
t
(3)
≥ min{r, γKSTlog(n)/2} ≥ γ log(n) , which concludes the proof of Theorem1 for ` = 3.
For general ` we proceed by induction and consider an `-partite graph G with vertex classes X1, . . . , X` of size n and cn` copies of K`. Similarly, as above we first locate a subset X10 ⊆ X1of size Ω(log n) such that the joint neighbourhood X20 of X10 in X2 is of size at least√
n and the subgraph induced on X10, X20, X3, . . . , X`
contains at least (c/2)|X12||X20|n`−2 copies of K`. Note that this time in the n- element set X1 we only located a subset of size Ω(log n).
However, in the same way as we located the subsets X00 in the set X0 in proof for ` = 3, we can find a subset X100 ⊆ X10 still of size Ω(log n) such that the joint neighbourhood X30 of X100 in X3 has size √
n and the subgraph induced on X100, X20, X30, X4, . . . , X` contains at least (c/4)|X100||X20||X30|n`−3 copies of K`.
We then repeat this argument ` − 3 more times and end up with sets X1(`−1)and X20, . . . , X`0 such that
• |X1(`−1)| = Ω(log n) and |X20|, . . . , |X`0| >√ n,
• Xi0⊆T
x∈X1(`−1)NG(x), and
• the number of copies of K`in the induced subgraph G[X1(`−1), X20, . . . , X`0] are at least (c/2`−1)|X1(`−1)||X20| · · · |X`0| .
In particular, G induced on X20, . . . , X`0contains at least (c/2`−1)|X20| · · · |X`0| copies of K`−1, which allows us to apply the induction assumption. Consequently, G
induced on X20, . . . , X`0 contains a t0-blow-up of K`−1 for t0 = Ω(log(√
n)), which together with X1(`−1)yields an Ω(log n)-blow-up of K` in G.
3. Large 3-partite 3-uniform hypergraphs in relatively dense hypergraphs
In this section we verify Theorem4 and discuss the proof of Corollary5.
Proof of Theorem 4. First we note that without loss of generality we may assume that G (and hence H) is a balanced tripartite (hyper)graph. In fact, for an n-vertex graph G = (V, E) consider the tripartite graph G0= (V0, E0) = G × K3 defined by V0 = V × [3] and {(x, i), (y, j)} is an edge in E0 if and only if xy ∈ E and i 6= j.
The graph G0 is tripartite and every triangle in G appears with all six different labelings in G0. Similarly we define H0 = H × K3(3) on the vertex set V0 with {(x, 1), (y, 2), (z, 3)} being an edge if and only if xyz ∈ E(H). Every edge of H appears six times in H0.
As a result we constructed a tripartite hypergraph H0 and a tripartite graph G0 with d(H0|G0) = d(H|G). Now a tripartite version of Theorem 4 yields a copy of Kt,t,t(3) in H0 with t ≥ K√
log n. This copy of Kt,t,t(3) may contain copies of the same vertex in more than one of the three partition classes. However, in each class we can select a set of at least t/3 vertices such that these collisions can be avoided.
Consequently, there is a copy of Ks,s,s(3) ⊆ H with s ≥ K√
log n/3, which suffices for this reduction and below we may assume that G and H are balanced tripartite.
Let K > 0 be given we fix β > 0 sufficiently small so that
− 4K2log(1 − 4β) < 1 . (5)
Let c > 0 be given. Without loss of generality we may assume that
0 < c ≤ β < 1/6 . (6)
Finally, we let n0 be sufficiently large such that for every n ≥ n0 we have Kp
log n + 1 ≤ nβ2c 8
(1 − 4β) βc
4
25/β3cK
√log n+1
(7) and
Kp
log n + 1 < n
β2c
8
4/βK
√log n+1
(1 − 4β)(K
√log n+1)2. (8)
Bounding K√
log n + 1 by 2K√
log n, we observe that in fact condition (8) holds for sufficiently large n due to the choice of β in (5).
Let G = (X ˙∪Y ˙∪Z, EG) be a tripartite graph with |X| = |Y | = |Z| = n and with at least cn3 triangles. Let V = V (G) = X ˙∪Y ˙∪Z and denote by K3(G) the set of triangles in G, i.e.,
K3(G) ={x, y, z} ⊆ V : xy, xz, and yz ∈ EG .
Let H ⊆ K3(G) be a 3-uniform hypergraph satisfying d(H|G) ≥ 1 − β. We set t =l
Kp log nm
(9) and will show that Kt,t,t(3) ⊆ H.
For the proof we also fix the following two auxiliary constants r > s > t s =
t
β(1 − 3β)
≤ 3t
β and r = 8s
β2c
≤ 25t
β3c. (10) For later reference we observe that the conditions on n0 in (7) and (8) yield
t ≤ 1
2(1 − 4β)tβ2c 4
βc
4
r
n (11)
and
t < β2c
8
s
(1 − 4β)tβ2c 8
t
n . (12)
The proof consists of three steps, each being rendered in one of the claims (Claims 1-3) below. We first state these claims and conclude the proof of The- orem4 from it, before we verify the claims.
Applying these claims succesively, we will be choosing sets X0, X00, and X000 with X ⊇ X0 ⊇ X00 ⊇ X000 and |X0| = r, |X00| = s, and |X000| = t so that a more and more refined structure can be deduced for the hypergraph restricted to those subsets and their neighbourhoods.
In the first step we select some special set of vertices X0⊆ X with |X0| = r and with “large” joint neighbourhood in Y , so that the induced graph and hypergraph on X0 it neighbourhood in Y and Z still enjoy similar properties, as G and H at the beginning (see(iv )and(iii )in the following claim).
Claim 1. There exist some sets X0 ⊆ X and Y0 ⊆ Y such that for the induced subgraph G0= G[X0, Y0, Z] and for H0 = H ∩ K3(G0) we have
(i ) |X0| = r and |Y0| ≥ (βc/4)rn, (ii ) Y0⊆ NG0(X0),
(iii ) d(H0|G0) ≥ 1 − 2β, and (iv ) |K3(G0)| ≥ βc|X0||Y0|n/2.
While Claim1 gives us some control over pairs (x, y) ∈ X0× Y0 the next claim achieves the same for pairs from X00× Z0 for suitably chosen subsets X00⊆ X0 and Z0 ⊆ Z.
Claim 2. There exist some sets X00 ⊆ X0 and Z0 ⊆ Z such that for the induced subgraph G00= G[X00, Y0, Z0] and for H00= H ∩ K3(G00) we have
(I) |X00| = s and |Z0| ≥ (β2c/8)sn, (II) Z0 ⊆ NG00(X00),
(III) d(H00|G00) ≥ 1 − 3β, and (IV) |K3(G00)| ≥ β2c|X00||Y0||Z0|/4.
The next step concerns the pairs between Y0 and Z0. We will select some subset X000 ⊆ X00 and a set of edges F ⊆ EG00(Y0, Z0) such that xyz ∈ E(H00) for every yz ∈ F and x ∈ X000.
Claim 3. There exist X000⊆ X00 and a set of edges F ⊆ EG00(Y0, Z0) such that (a ) |X000| = t and |F | ≥ (1 − 4β)tβ2c|Y0||Z0|/4 and
(b ) xyz ∈ E(H) for every x ∈ X000 and yz ∈ F .
One can check that the bounds for the sizes of Y0, Z0and F (see(i )of Claim1,(I) of Claim2, and(a )of Claim 3) combined with (11) and (12) imply
(t − 1)|Y0| t
< |Z0||F |/|Z0| t
.
Consequently, Theorem 6 yields a copy of Kt,t in the bipartite graph (Y0∪Z˙ 0, F ).
This copy of Kt,ttogether with X000spans a copy of Kt,t,t(3) in H. This concludes the
proof of Theorem4.
Proof of Claim1. In the proof the following obvious identities will be used. If uv is an edge in G, then |NG(u, v)| is the number of triangles containing this edge and
|K3(G)| = X
xy∈EG(X,Y )
|NG(x, y)|
= X
xz∈EG(X,Z)
|NG(x, z)| = X
yz∈EG(Y,Z)
|NG(y, z)| .
(13)
Moreover, for an edge uv ∈ EG we denote by NH(u, v) the set of those vertices w ∈ V for which uvw is a hyperedge of H. Clearly,
|H| = X
xy∈EG(X,Y )
|NH(x, y)| = X
xz∈EG(X,Z)
|NH(x, z)| = X
yz∈EG(Y,Z)
|NH(y, z)| (14)
and since H ⊆ K3(G) we have
NH(u, v) ⊆ NG(u, v) .
First we remove every edge xy from EG(X, Y ) for which either (a ) |NH(x, y)| < (1 − 2β)|NG(x, y)| or
(b ) |NG(x, y)| < βcn/2.
Let Ga be the tripartite subgraph of G, which we obtain after removal of all edges from EG(X, Y ) due to reason(a )and set Ha = H ∩ K3(Ga).
Owing to the identities (13) and (14) we have
|E(H) \ E(Ha)| = X
xy6∈EGa(X,Y )
|NH(x, y)|
(a )
≤ X
xy6∈EGa(X,Y )
(1 − 2β)|NG(x, y)|
≤ (1 − 2β) X
xy∈EG(X,Y )
|NG(x, y)| = (1 − 2β)|K3(G)|
Moreover, since |E(H)| ≥ (1 − β)|K3(G)| by assumption of Theorem4 we have
|K3(Ga)| ≥ |E(Ha)| = |E(H)| − |E(H) \ E(Ha)| ≥ β|K3(G)| ≥ βcn3. Next we remove the edges from EGa(X, Y ) which fail to satisfy condition(b )and let G0 be the resulting graph and set H0 = H ∩ K3(G0). Since eGa(X, Y ) ≤ n2 we destroy at most βcn3/2 triangles this way. Consequently,
|K3(G0)| ≥ |K3(Ga)| − βcn3/2 ≥ βcn3− βcn3/2 = βcn3/2
and, therefore, eG0(X, Y ) ≥ βcn2/2. Note that every edge xy in EG0(X, Y ), then satisfies conditions(a )and(b ).
A simple double counting argument as was carried out by K¨ovari, S´os, and Tur´an in the proof of Theorem6will yield a complete bipartite subgraph in G0[X, Y ] with the desired properties. In fact,
X
y∈Y
|NG0(y) ∩ X|
r
≥ neG0(X, Y )/n r
≥ nβcn/2 r
≥ n βc 4
r
n r
,
where we used βcn/4 ≥ r (see (10) and (11)) for the last inequality.
Hence, there exists an r-element subset X0⊆ X with |NG0(X0)| ≥ (βc/4)rn and we set Y0 = NG0(X0). By definition X0 and Y0 satisfy properties (i ) and (ii ) of Claim1. Moreover, properties(iii )and (iv )follow from properties(a )and(b )of
the graph G0.
The proof of Claim2is almost identical to the proof of Claim1.
Proof of Claim2. Let G0 and H0 be given by Claim1. We remove every edge xz from EG0(X0, Z) for which
(c ) |NH0(x, z)| ≥ (1 − 3β)|NG0(x, z)| or (d ) |NG0(x, z)| ≥ β2c|Y0|/4.
Let Gc be the tripartite subgraph of G0, which we obtain after we remove all edges from EG0(X0, Z) which violate condition (c ) and let Hc = H ∩ K3(Gc). Owing to (iii ) of Claim 1 we have |H0| ≥ (1 − 2β)|K3(G0)|. Similarly as in the proof of Claim1one can verify that |H0\ Hc| ≤ (1 − 3β)|K3(G0)| and, hence, we obtain
|K3(Gc)| ≥ |Hc| ≥ β|K3(G0)| ≥ β2c|X0||Y0|n/2 ,
where the last inequality follows from (iv ) of Claim 1. Now delete those edges from EGc(X0, Z) which fail to satisfy (d ). Let G00 be the resulting graph and set H00 = H ∩ K3(G00). Since eGc(X0, Z) ≤ |X0|n we destroy at most β2c|X0||Y0|n/4 triangles this way. Consequently,
|K3(G00)| ≥ |K3(Gc)| − β2c|X0||Y0|n/4 ≥ β2c|X0||Y0|n/4
and, therefore, eG00(X0, Z) ≥ β2c|X0|n/4. By definition of G00 every edge xz in EG00(X0, Z) satisfies conditions(c )and(d ).
A similar double counting argument as in the proof of Claim1yields X
z∈Z
|NG00(z) ∩ X0| s
≥ neG00(X0, Z)/n s
≥ nβ2c|X0|/4 s
≥ n β2c 8
s|X0| s
,
where we used β2c|X0|/8 = β2cr/8 ≥ s (see (10)) for the last inequality. There- fore, there exists an s-element set X00 ⊆ X0 with |NG00(X00)| ≥ (β2c/8)sn. We set Z0 = NG00(X00). By definition X00 and Z0 satisfy properties (I) and (II) of Claim2. Moreover, properties(III)and(IV)follow from properties(c )and(d )of
the graph G00.
Proof of Claim3. Owing to part (ii ) of Claim 1 and to part (II) of Claim 2 the two bipartite graphs G00[X00, Y0] ⊆ G0[X0, Y0] and G00[X00, Z0] are complete. Con- sequently, every edge yz ∈ EG00(Y, Z) belongs to exactly |X00| triangles in G00. Hence property(III)of Claim2implies that on average an edge yz ∈ EG00(Y, Z) is
contained in (1 − 3β)|X00| hyperedges of H00. Consequently, X
yz∈EG00(Y,Z)
NH00(y, z) t
≥ eG00(Y, Z)(1 − 3β)|X00| t
≥ (1 − 4β)teG00(Y, Z)|X00| t
,
where we used (1 − 3β)|X00| − t ≥ (1 − 4β)|X00|, which follows since β|X00| = βs ≥ t by (10). Hence, there exists a t-set X000⊆ X00such that at least (1 − 4β)teG00(Y, Z) of the edges of EG00(Y, Z) form a hyperedge with every x ∈ X000. It follows from(IV) of Claim2 that eG00(Y, Z) ≥ β2c|Y0||Z0|/4. Consequently, there exists a subset F of EG00(Y, Z) of size at least
(1 − 4β)teG00(Y, Z) ≥ (1 − 4β)tβ2c|Y0||Z0|/4 ,
which satisfies property(b )of Claim3.
Below we sketch a proof of Corollary5 based on Theorem4 and the regularity method for hypergraphs.
Proof of Corollary5(sketch). Let a 3-uniform hypergraph F and a constant K > 0 be given. Consider an n-vertex, 3-uniform hypergraph H for sufficiently large n.
We assume that neither H nor its complement contain an s-blow-up Ks,s,s(3) of a single edge K3(3) for any s ≥ K√
log n. By Theorem4 this means that there exists a constant β > 0 such that for any c > 0 and sufficiently large n the following holds:
If G is a graph on the same vertex set as H and G contains at least cn3 triangles, then
β < d(H|G) < 1 − β . (15)
We will use the hypergraph regularity lemma from [9], which yields some auxiliary constants t and ` which can be bounded from above in terms of δ (independent of n and H). In particular, we can ensure the following hierarchy of the constants involved in this proof
1 K, 1
|V (F )| β δ 1
`,1
t c 1
n. (16)
Below we will use (15) to infer that H contains an induced copy of F .
To this end we apply the regularity lemma from [9]. We will not describe the concept of regularity of that lemma here, but we will discuss its consequences necessary for the proof. This regularity lemma decomposes the vertex set of H into t vertex classes V1, . . . , Vtof size bn/tc or dn/te each and it partitions each of the
t
2 complete bipartite graphs K(Vi, Vj) into ` bipartite graphs Pijαsuch that for all but at most δ 3t`3 of the tripartite graphs (so-called triads)
Pijkαijαikαjk= (Vi∪V˙ j∪V˙ k, Pijαij∪P˙ ikαik∪P˙ jkαjk) for 1 ≤ i < j < k ≤ t and 1 ≤ αij, αij, αjk≤ ` we have
(a ) Pijkαijαikαjk contains at least 12(`tn)3 triangles and (b ) H is δ-regular with respect to Pijkαβγ.
Since almost all triads satisfy properties(a )and(b )a standard averaging argument (see, e.g., [13]) yields a collection of indicies 1 ≤ i1≤ · · · ≤ if ≤ t where f = |V (F )|
and αii,ij ∈ [`] for all 1 ≤ i < j ≤ f such that properties(a )and (b )hold for all triads given by this selection, i.e., for all f3 triads of the form
Piαiiijαiiikαij ik
iijik .
It follows from (15), (16), and property (a ) that for all those selected triads P satisfy β < d(H|P ) < 1 − β. Morover, the regularity lemma also asserts that the involved bipartite graph Pijα are sufficiently regular themselves. This allows us to apply the counting lemma from [14] (see also [11]) in this environment, which then
yields an induced copy of F .
4. A few results towards Problem2
In view of Erd˝os’ result (see (1)) Problem 2 is true for all r-partite, r-uniform hypergraphs F . In fact, if F has ` vertices and an n-vertex r-uniform hypergraph H contains Ω(n`) copies of F , then it must contain at least Ω(nr) hyperedges. Hence, it follows from (1) that H contains a t-blow-up of Kr(r) for some t = Ω((log n)
1 r−1) and such a t-blow-up contains trivially a (t/`)-blow-up of F .
4.1. A non-partite example. We solve Problem2for the following non-partite r- uniform hypergraph Sr. The hypergraph Srhas vertex set {x1, . . . , xr−1, y1, . . . , yr} and it contains the edge {y1, . . . , yr} and the r edges of the form {x1, . . . , xr−1, yj} with j ∈ [r].
Proposition 7. For every integer r ≥ 3 and every c > 0 there exist γ > 0 and n0
such that every r-uniform hypergraph H on n ≥ n0 vertices which contains at least cn2r−1 copies of Sr must contain a t-blow-up of Sr for some t ≥ γ(log n)r−11 .
The proof of Proposition7relies on Lemma8, which is variant of Erd˝os’ result (see also [16, Theorem 2]). For an r-uniform hypergraph H and an (r − 1)-tuple
~v = (v1, . . . , vr−1) of its vertices we denote by dH(~v) its degree in H, i.e., the number of edges in H which contain v1, . . . , vr−1.
Lemma 8. For every integer r ≥ 3 and every c1, c2 > 0, and C > 0 there exist γ > 0, D > 0, and n0 such that for every n ≥ n0 the following holds. Let H be an r-partite, r-uniform hypergraph with vertex classes V1, . . . , Vr such that
|V1| = · · · = |Vr−1| = n and |Vr| = m ≥ c1(log n)r−11 . Let F ⊆ {~v ∈ V1× · · · × Vr−1: dH(~v) ≥ c2m} . If
|F | ≥ nr−1 exp(C(log n)r−11 )
, then there exist subsets F0 ⊆ F and W ⊆ Vr such that
(i ) |W | = γ(log n)r−11 and |F0| ≥ nr−1exp(−D(log n)r−11 ) and
(ii ) for every (v1, . . . , vr−1) ∈ F0 and every w ∈ W we have {v1, . . . , vr−1, w}
is an hyperedge in H.
Proof. Since dH(~v) ≥ c2m for every ~v ∈ F we have X
~ v∈F
dH(~v) t
≥ |F |c2m t
≥c2
2
t
|F |m t
for every integer t ≤ c2m/2. Consequently, there exist a subset W ⊆ Vr with
|W | = t and a subset F0⊆ F of size at least (c2/2)t|F | for which(ii )holds. Setting t =jc1c2
2 (log n)r−11 k
≤ c2
2m and in view of
|F0| ≥c2 2
t
|F | ≥ nr−1
exp
(C + ln(2/c2)c1c2/2)(log n)r−11 , and
|W | = t ≥ c1c2
3 (log n)
1 r−1
we infer the lemma for γ = c1c2/3, D = (C + ln(2/c2)c1c2/2), and sufficiently
large n.
We obtain Proposition7from Lemma8.
Proof of Proposition7. Similarly as in the proofs of Theorems 1 and 4, we note that it suffices to prove a partite version of Proposition 7. More precisely, we will assume that H is contained in an n-blow-up of Sr. Let X1, . . . , Xr−1, Y1, . . . , Yr be the vertex classes of H. The proof splits into three parts.
In the first step we consider only those hyperedges E0of H[Y1, . . . , Yr] with the property, that each of them is contained in at least cnr−1/2 copies of Sr. Since this way we ignore at most cnr−1/2 copies of Srin H, there are at least cnr−1/2 copies left. In particular, |E0| ≥ cnr/2 and by an application of Erd˝os result (see (1)) we locate in E0 a t0-blow-up of Kr(r) for t0 = γ0(log n)r−11 for some γ0 = γ0(c). Let Y10, . . . , Yr0 be the vertex classes of this t0-blow-up.
In the second step we consider the induced r-uniform subhypergraph H1 = H[X1, . . . , Xr−1, Y10, . . . , Yr0]. Owing to the definition of E0 the hypergraph H1
contains at least (c/2)nr−1|Y10| . . . |Yr0| copies of Sr.
Consequently, there exists a set F0⊆ X1× · · · × Xr−1 of size at least (c/4)nr−1 such that for every i ∈ [r] every (x1, . . . , xr−1) ∈ F0 there exist at least (c/4)|Yi0| vertices y ∈ Yi0 such that {x1, . . . , xr−1, y} is an edge in H1.
The second step consists of r applications of Lemma 8. We will successively select sets F1⊆ F0 and Y100⊆ Y10, then F2⊆ F1 and Y200⊆ Y20 and so on such that for every j = 0, 1, . . . , r the set Fj is still “large” and, moreover, for every (r − 1)- tuple (x1, . . . , xr−1) ∈ Fj and every y ∈ Y100∪ · · · ∪ Yj00the r-tuple {x1, . . . , xr−1, y}
forms an edge in H1.
For the first application we apply it to the hypergraph H1[X1, . . . , Xr−1, Y10] and the set F0. Lemma 8 yields subsets Y100 ⊆ Y10 and F1 ⊆ F0 satisfying for some constants γ1= γ1(c, γ0) = γ1(c) > 0 and C1= C1(c, γ1)
(i ) |Y100| = γ1(log n)r−11 and |F1| ≥ nr−1exp(−C1(log n)r−11 ) and
(ii ) for every (x1, . . . , xr−1) ∈ F1 and every y ∈ Y100 we have {x1, . . . , xr−1, y}
is an hyperedge in H1.
We continue with the second application of Lemma8. For that we note that since by definition of F0 ⊇ F1 every ~x ∈ F1 extends to (c/4)|Y20| edges in H with the r-th vertex being a member of Y20. Consequently, we can apply Lemma8again and obtain subsets Y200⊆ Y20 and F2⊆ F1 such that
(i ) |Y200| = γ2(log n)r−11 and |F2| ≥ nr−1exp(−C2(log n)r−11 ) and
(ii ) for every (x1, . . . , xr−1) ∈ F2 and every y ∈ Y200 we have {x1, . . . , xr−1, y}
is an hyperedge in H1
for some constants γ2 and C2only depending on c.
We repeat this procedure r−2 more times and obtain subsets Yj00for every j ∈ [r]
and F0⊇ F1⊇ F2⊇ · · · ⊆ Frsuch that for every j ∈ [r] we have (i ) |Yj00| = γj(log n)r−11 and |Fj| ≥ nr−1exp(−Cj(log n)r−11 ) and
(ii ) for every (x1, . . . , xr−1) ∈ Fj and every y ∈ Yj00 we have {x1, . . . , xr−1, y}
is an hyperedge in H1
for some constants γj and Cj only depending on c.
In the third and last step we consider the set Fr, which we view as an (r − 1)- partite, (r − 1)-uniform hypergraph with vertex classes X1, . . . , Xr−1. Since
|Fr| ≥ nr−1 exp(Cr(log n)r−11 )
≥ nr−1 n1/sr−2
for some integer s ≥ γ(log n)r−11 with γ > 0 only depending on Cr, which only depends on the given c. Hence, it follows from Erd˝os’ result [4, Theorem 1] that Fr contains a copy of a s-blow-up of Kr−1(r−1). Owing to (ii ) for every j ∈ [r] this s-blow-up together with the sets Y100, . . . , Yr00spans a t-blow-up of Srfor the integer
t = min{t1, . . . , tr, s}.
4.2. Partite blow-ups. As discussed in Section4.1for r-partite r-uniform hyper- graphs F a positive answer for Problem2is a direct consequence of (1). However, for partite blow-ups (defined below) the problem is different. In this section we address such a version of the problem for tight 3-uniform paths.
A tight path P`(3) on vertex set [`] = {1, . . . , `} consists of all “consecutive”
triplets of vertices, i.e., e ∈ E(P`(3)) if and only if e = {i, i + 1, i + 2} for some i = 1, . . . , ` − 2. Moreover, we say an `-partite hypergraph H with vertex classes V1, . . . , V` contains a partite copy of an `-vertex hypergraph F with V (F ) = [`] if the copy of the vertex i ∈ V (F ) is contained in Vi. We will also say that H contains a partite blow-up of F , if it contains a blow-up with vertex classes U1, . . . , U` such that Ui⊆ Vi for every i ∈ V (F ).
Proposition 9. For all integers ` ≥ 3 and every c > 0 there exists γ > 0 and n0
such that the following holds. If a 3-uniform, `-partite hypergraph H is contained in an n-blow-up of the tight path P`(3) and it contains at least cn` partite copies of P`(3), then H contains a partite t-blow-up of P`(3) with t ≥ γ√
log n.
The proof of Proposition 9 relies on the following version for 3-uniform hyper- graphs of the result of Erd˝os from [4].
Lemma 10. For every c ∈ (0, 1) there exist γ > 0 and n0such that for every n ≥ n0 the following holds. Let H = (X ˙∪Y ˙∪Z, E) be a tripartite 3-uniform hypergraph with
|X| ≥ (c/8)√
log n, |Y | ≥√
n, and |Z| = n. If every pair (x, y) ∈ X × Y satisfies dH(x, y) ≥ cn, then there exist subsets X0⊆ X, Y0⊆ Y , and Z0⊆ Z such that
(a ) |X0| ≥ γ√
log n, |Y0| ≥ (c/8)√
log n, and |Z0| ≥√ n and (b ) H[X0, Y0, Z0] is a complete, tripartite, 3-uniform hypergraph.
Proof. The proof relies on two applications of the K¨ovari-S´os-Tur´an theorem. We set
γ = c2/20 .
Let n be sufficiently large and set r = dγp
log ne < c|X|/2 , s = d(c/8)p
log ne , and t = d√
ne . (17) Owing to the minimum degree condition for the pairs in X × Y we have the lower bound c|X| for the average degree of the pairs in Y × Z. Consequently, we have
X
(y,z)∈Y ×Z
dH(y, z) r
≥ |Y ||Z|c|X|
r
≥c 2
r
|Y ||Z||X|
r
.
Hence, in view of γ < 1 there exist sets X0⊆ X of size r and F ⊆ Y × Z of size
|F | ≥c 2
r
|Y ||Z| ≥c 2
√log n
|Y ||Z| (18)
such that for every (y, z) ∈ F and x ∈ X0 we have that {x, y, z} is an edge of H.
Next we apply Theorem6 to the auxiliary bipartite graph G with vertex parti- tion Y ˙∪Z and edge set F . Note that due to the assumptions on the sizes of Y and Z and due to (18) it follows from the choice of s and t in (17) that for sufficiently large n we have
(t − 1)|Y | s
< |Z|c 4
s
√log n|Y | s
< |Z||F |/|Z|
s
,
since (c/8) log(4/c) < 1/2 for every c ∈ (0, 1). Hence, Theorem6 yields a copy of Ks,tin G. Set Y0⊆ Y be the vertex class of size s and Z0⊆ Z be the vertex class of size t of this copy of Ks,t. It then follows that X0, Y0 and Z0 satisfy the conclusion
of the lemma.
Proposition9 is a consequence of ` − 2 applications of Lemma10.
Proof of Proposition9. For given ` ≥ 3 and c > 0 let γ > 0 and n0be the constants guaranteed by Lemma10applied with ` and
c0= c 2(` − 2).
Let n ≥ n0 be sufficiently large and let H be a 3-uniform hypergraph which is contained in an n-blow-up of the tight path P`(3) with ` vertices and which itself contains at least cn` partite copies of P`(3). Let V1, . . . , V` be the vertex classes of H.
We first remove all hyperedges from H which contain a pair which fails to satisfy the following degree condition. For i = 1, . . . , ` − 2 and a pair of vertices (vi, vi+1) ∈ Vi× Vi+1 we denote by dH,i+2(vi, vi+1) the number of neighbours of those two vertices in Vi+2, i.e.,
dH,i+2(vi, vi+1) =
v ∈ Vi+2: {vi, vi+1, v} ∈ E(H) .
Successively, we will keep removing hyperedges {vi, vi+1, v} from H with vi ∈ Vi, vi+1 ∈ Vi+1and v ∈ Vi−1∪Vi+2for some i = 1, . . . , `−2 as long as dH,i+2(vi, vi+1) <
c0n. Let H1 be the hypergraph of the remaining edges. By definition H1 has the property, that if a pair (vi, vi+1) ∈ Vi× Vi+1 is contained in some edge of H1, then
dH,i+2(vi, vi+1) ≥ c0n .
Note that since there are ` − 2 choices for i, we removed less than (` − 2)n2× c0n × n`−3≤ cn`/2
copies of P`(3)from H this way. Hence, there are at least cn`/2 partite copies of P`(3) in H1left and, consequently, H1[V1, V2, V3] contains at least cn3/2 hyperedges. Let F be the set of those pairs in V1× V2which are contained in at least one (and hence in at least c0n) hyperedges in H1[V1, V2, V3]. Clearly, |F | ≥ cn2/2 and, moreover, for sufficiently large n we have
√n
|V1|
(c0/8)√ log n
< n (1/2)cn2/|V2|
(c0/8)√ log n
.
Consequently, Theorem 6 yields the existence of a copy of Ks,t in F for integers s ≥ (c0/8)√
log n and t ≥√
n. Let V10 and V20 be the corresponding vertex sets in V1and V2 with sizes |V10| = s and |V20| = t.
Since every pair of (v1, v2) ∈ F was contained in an hyperedge of H1, we have dH,3(v1, v2) ≥ c0n. Therefore, we can apply Lemma 10 and obtain vertex sets W1⊆ V10, V200⊆ V20 and V30⊆ V3 of sizes
|W1| ≥ γp
log n , |V200| ≥ (c0/8)p
log n , and |V30| ≥√ n
such that H1[W1, V200, V30] is a complete tripartite hypergraph. In particular, every pair in (v2, v3) ∈ V200× V30 is contained in an edge of H1. Hence by definition of H1, dH,4(v2, v3) ≥ c0n. In other words H1[V200, V30, V4] meets the assumptions of Lemma10for X = V200, Y = V30, and Z = V4. Therefore, there exist sets W2⊆ V200, V300⊆ V30 and V40⊆ V4of sizes
|W2| ≥ γp
log n , |V300| ≥ (c0/8)p
log n , and |V40| ≥√ n such that H1[W2, V300, V40] is a complete tripartite hypergraph.
We repeat the same argument for every i = 3, . . . , `−2 and obtain sets W3, . . . , W`−2
and V`−100 and V`0. If follows that H1[W1, . . . , W`−2, V`−100 , V`0] contains a t-blow-up of P`(3) for t ≥ γ√
log n.
5. Concluding remarks
Much of the work here was motivated by Problem 2. We point out that a somewhat weaker conjecture was formulated by Bollob´as, Erd˝os and Simonovits in [1]. For an r-uniform hypergraph F let πF≥ 0 denote its Tur´an density defined by
lim
n→∞
max{|E(H)| : H is an F -free n-vertex r-uniform hypergraph}
n r
.
It is not hard to show that this limit in fact exists (see, e.g., [10]). Moreover, it is known (see, e.g., [7]) that for fixed ε > 0 and sufficiently large n every n-vertex r-uniform hypergraph with at least (πF + ε) nr
edges must contain cn` copies of F , where ` = |F | and c = c(ε) > 0 only depends on ε. In other words, such hypergraphs H satisfy the assumptions of Problem2and, in fact, it was conjectured in [1] that those satisfy the conclusion of Problem2. (In [1] such a result was proved for graphs.)
Our contributions here rely mainly on applications of the results of K¨ovari, S´os, and Tur´an [12] and Erd˝os [4] and very similar ideas. We believe those ideas may be used to approach Problem 2 for other particular hypergraphs F . However, for major progress on this problem new ideas seem to be needed.
